Wald for non-stopping times: The rewards of impatient prophets

Let X1, X2, . . . be independent identically distributed nonnegative random variables. Wald’s identity states that the random sum ST := X1 + · · · + XT has expectation ET ·EX1 provided T is a stopping time. We prove here that for any 1 < α ≤ 2, if T is an arbitrary nonnegative random variable, then ST has finite expectation provided that X1 has finite α-moment and T has finite 1/(α− 1)-moment. We also prove a variant in which T is assumed to have a finite exponential moment. These moment conditions are sharp in the sense that for any i.i.d. sequence Xi violating them, there is a T satisfying the given condition for which ST (and, in fact, XT ) has infinite expectation. An interpretation is given in terms of a prophet being more rewarded than a gambler when a certain impatience restriction is imposed.